Instead, Spurs suffered a 2-0 defeat to their Premier League rivals at the Wanda Metropolitano Stadium in Madrid.
As a result, Pochettino said he did not think it was a good time to step away from the job he has held since 2014, per the Press Association (h/t Sporting Life):
"Maybe if it was a different result after the final you can think, 'OK, maybe this is a moment to step out of the club, leave the club and give them the possibility of a really new chapter with a new coaching staff'. But after the final I felt this was not great to finish like this. I'm not a person that avoids facing problems or a difficult situation.
"I love a massive challenge, a difficult challenge, and of course now to rebuild that mentality to make it possible to repeat a similar season. That is exciting and motivates me a lot.
"For sure. You know very much when you touch glory you behave differently or you feel different or the players feel different, and the challenge becomes different."
Pochettino, 47, has done a remarkable job at Spurs.
He has made them perennial Champions League qualifiers. Before he arrived at the club, Tottenham had finished in the Premier League's top four just twice.
They have now had four top-four finishes in succession and will be strongly placed to do it again in 2019-20, not least as they have finally made some new signings.
The Spurs squad that reached a first-ever Champions League final in 2018-19 was the same Pochettino had the season before.
Tottenham did not make a single signing last summer or in January.
Already this summer, though, they have made the key addition of midfielder Tanguy Ndombele, who signed from Lyon for £55 million earlier this month:
Sky Sports Statto @SkySportsStatto
📊 Tanguy Ndombele league stats last season compared to @SpursOfficial players Most successful dribbles 6️⃣4️⃣ NDOMBELE 4️⃣8️⃣ Son 4️⃣5️⃣ Lucas Moura Most assists 1️⃣2️⃣ Eriksen 7️⃣ NDOMBELE 6️⃣ Son League assists in last 2 seasons 1️⃣3️⃣ NDOMBELE 1️⃣3️⃣ Dele Alli https://t.co/NGDpf0BziS
Pochettino has managed to make Spurs into Champions League finalists working on a shoestring budget.
If Spurs make more investment before the start of the 2019-20 season, the north London club could be in for an even better campaign, and they will benefit from having the Argentinian in charge still.